Tag: Problem-solving

This post is contributed by Simon Clay who is part of the Teacher Support team at MEI.

Given that the changes to A level mathematics are significant, an overhaul of teaching schemes for the new two-year long qualification is not a trivial task. During 2016-17 a number of members of staff at MEI developed a Scheme of Work for the new A levels with the aim of trying to produce something useful for as wide a range of audience as possible. This result is this freely available SoW, accessible via the MEI website.

Some of the thinking behind the design of the SoW units was as follows:

– It aimed to break down the new A level content into manageable units.

– It needed to function as a starting point for discussions in departments and therefore needed to be editable.

– It needed to take seriously the changes in emphasis of the new A levels, including the three overarching themes – Mathematical argument, language and proof; Mathematical problem solving; Mathematical modelling.

– It needed to incorporate useful features such as ideas as to how the use of technology can permeate the teaching of A level mathematics, questions which promote mathematical thinking, etc.

– It needed to be both adaptable and useable in the classroom.

– It needed to exemplify, and give free access to, some high quality teaching resources which can be ‘picked up and used’ in any classroom.

Since its launch in March, we have been pleased with the way the SoW has been received. A common request, however, was for the provision of a plan for how the units could be linked together in a cohesive way to ensure the content is covered in the time available. We have therefore worked on producing a series of schedules which show how the units of the first year (or AS content) can be arranged depending on considerations or constraints a department may have e.g. two teachers sharing a group, one of whom teaches pure and mechanics while the other teaches pure and statistics. (We have so far only tackled Year 1 content but Year 2 will follow in due course!)

The reason for a post in this blog is because Schedule E is as a result of the thinking and work done by Bruce and the team at TGA Redditch. It has been my privilege to take part in the discussions in which this SoW Schedule has been developed.

Below is an image of Schedule E taken from mei.org.uk/2017-sow and beneath this I describe the key features:

– The team wanted to begin the course with an emphasis on problem-solving and proof in order to set the culture of working in this way from lesson 1. This means lesson 1 will contain no mathematics beyond GCSE and will instead focus on reasoning, language and proof. Lesson 2 will look at indices but with an emphasis on reasoning and proof rather than subject content coverage.

– There was a strong desire to get the students working with and becoming familiar with the large data set (LDS) right from the start of the course. Thus by the end of the first teaching week students will know about the LDS and have done some initial exploratory work using it.

– The team identified some units, namely ‘Problem-solving’ and ‘Graphs and transformations’ as being recurring themes which can be addressed in a number of different units throughout the course rather than taught as discrete topics.

– The team wanted to use a teaching model where the class is shared between two members of staff but essentially runs as a single series of lessons. This will clearly involve a high level of collaboration between them but they are keen to dovetail their teaching so that the student experience is as coherent and seamless as possible.

– They wanted the applied units to be taught alongside the relevant pure unit so it is clear what mathematics is being applied. It is hoped that this will also help with fitting in the content in the time available.

– They wanted technology to be used by teachers and students whenever possible and so in the first few weeks there are planned opportunities for this, in particular when analysing the LDS and exploring graphs of exponential functions .

– The school has made a central decision that all students need to be prepared and entered for AS level examinations at the end of Year 12. This means that although at points it would be nice to extend and cover Year 2 topics straightaway these will need to wait.

And now there are only a few weeks until the schedule can be implemented!

In our work on revamping the curriculum for the new specification we have been careful to make sure that we are considering the overarching themes of problem solving and use of technology. We are very keen to ensure that use of technology does not become ever more complicated and ‘interactive’ PowerPoint files that are demonstrated from the front of the class room with little chance for students to use and develop their own skills. We also want to introduce this aspect as early as possible to encourage students to think about technology as a vehicle for working on and solving problems when they get stuck. On Friday I met with Simon, Fiona Kitchen (from FMSP) and two colleagues from my department to discuss methods for this.

Our starting point was to use the worksheet “Problem Solving with Geogebra” from MEI’s scheme of work. We looked at solving the problems ourselves, trying to limit techniques to those that year 12 students were able to use. This proved rather difficult! After much wrestling (and a plea to twitter) we managed to create working models in Geogebra for the first three of the problems.

We had a lot of fun working on these problems but concluded that the level was a too high for students who are starting out in year 12. As such we will need to adapt to something closer to GCSE if we are going to introduce Geogebra in this way at the beginning of the course. One thing that struck me during the afternoon was that we persevered with the problems for a long period, around 2.5 hours. This was something that our students would have really struggled to achieve. The same morning one of my year 11 students, when confronted with a difficult question, said “It’s alright for you sir, you are good at maths and can do it easily.” I was unable to make her understand that I don’t find all maths easy and that I enjoy the struggle with harder problems. This perhaps sums up the major problem we have been fighting against with our A-level students over the last few years – the lack of resilience as soon as a problem gets complicated.

Hopefully this process of really concentrating on both problem solving and technology will help to improve this, which is certainly the focus of what we are looking at during this process. For now though here are the problems that we worked on.

Problem 1

With this problem I found it very easy to create a polynomial fixed by the points A, B and C. This initially created a point D that moved as A, B and C moved. To do this I used the measuring tool in Geogebra to calculate how far away the points were from the origin.

My second attempt used a division at the start of the polynomial to allow me to control point D as well. It was at this point that I realised the shortcomings of my method of measuring distances – when I moved the points to negative values Geogebra continued to measure them as positive. To complete the problem Simon showed me how to use just the x-ordinate of A etc. in the calculation. My final solution is at: https://www.geogebra.org/m/bZD6fARj

Problem 2

For this problem I started by drawing a circle centred on the origin with a point on the circumference fixed into the side AC. I then created another circle centred on C which connected to the first. I repeated this methodology to create a third circle centred on point B. The three circles could then be manipulated together until I had a solution that worked. This did not satisfy me – I wanted to be able to change the triangle and the circles to remain a solution to the puzzle.

My instincts for this puzzle were probably from spending time playing the mobile phone game Euclidea – maybe those hours were not completely wasted! I guessed that the points where the circles met on the edges of the triangle were those where the circle inscribed in the triangle also touched the sides of the triangle. My resulting solution can be found here: https://www.geogebra.org/m/jMSThZg8

Problem 3

Problem 3 caused me the most problems. For a long time I was able to either create a line that was perpendicular to the tangent from A or a line that passed through the point B but not both. After a long time trying (and a plea to Twitter), Simon came up with a solution that can be found at: https://www.geogebra.org/m/MFgcJaAd

Problem 4

We ran out of time before tackling problem 4 and I have not yet returned to it. I have some ideas about using a quadratic function with roots that are the x-ordinates of A and B and integrating it but have not progressed any further yet. I leave that one with you…

Today I attended an MEI CPD event on the “Principles of the New A-Level.” I came away with a lot of ideas and things to think about further as I continue to work towards starting teaching the new syllabus in September. One of the new “Overarching Themes” in the specification for A-level maths is problem solving and it was this that my thoughts have kept returning to this evening. What is problem solving in mathematics, and specifically what is it that makes on question a problem solving task?

Take the picture here (one of the tasks we worked on at the session). What might students be asked to do? As an example we might ask “find the area of the rectangle”. In the previous specification students were likely to be guided to a solution, with parts of a question asking them to find relevant, intermediate, pieces of information such as the equation of the line that joins A and D. The idea of removing this and having to think really appeals to me as a mathematician but scares me as a teacher.

What interested me most about this was my own response to then thinking about how I would approach the question. I initially decided that I should work out the coordinates of the points A, B and D, then work out the lengths of segments AD and AB and multiply them together to find the area of the rectangle. As I started working through the problem it quickly dawned on me that having worked out A, B and D it was much more efficient to work out the area of the triangle made by them, and then double it, rather than using Pythagoras theorem twice. While my initial strategy would have led to a correct solution, my adaptation of it led to an easier process. However, if I had not been able to think of a strategy at all I would not have been able to start the problem.

Another aspect of the new schemes is the use of technology. Some of these problem solving tasks are better explored with dynamic software. We looked at a geometric method for showing that sin (a + b) = sin a cos b + cos a sin b. This looked at drawing a rectangle and dividing the area up in different ways. One area students may struggle to grasp is that this is true for all values of a and b. By creating a dynamic model we are able to illustrate that it holds for different values and we can then generalise.

We are going to have to make sure that we build enough of these sorts of tasks into our lessons so that students have experience of the thought processes required to answer them. This will obviously require time to allow students to develop their skills. Something that I am currently looking at to help free up time in the classroom is flipped learning, but that is a topic for another day.

I am off to try and make a geogebra applet that shows sin (a + b) = sin a cos b + cos a sin b!

[My first attempt at the geogebra is available here: https://www.geogebra.org/worksheet/edit/id/qtNKm2pa Needs some work on the labelling!]